Counterexamples to maximal regularity for operators in divergence form

2024/01/10 by Bechtel, Sebastian, Mooney, Connor, Veraar, Mark
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary: 35K90 #Secondary: 35B65

paper · doi:10.48550/arxiv.2401.05550

Abstract

In this paper, we present counterexamples to maximal L^p-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions' theory that such operators admit maximal L²-regularity on H^-1 under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal L^p-regularity on H^-1(ℝ^d) or L²-regularity on L²(ℝ^d).

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